Oscillation of second-order differential equations with a sublinear neutral term

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1 CARPATHIAN J. ATH ), No. 1, 1-6 Online version available at Print Edition: ISSN Online Edition: ISSN Oscillation of second-order differential equations with a sublinear neutral term RAVI P. AGARWAL, ARTIN BOHNER, TONGXING LI and CHENGHUI ZHANG ABSTRACT. This paper is concerned with oscillation of a certain class of second-order differential equations with a sublinear neutral term. Two oscillation criteria and two illustrative examples are included. In particular, the results obtained improve those reported in the literature. 1. INTRODUCTION The neutral differential equations find numerous applications in natural sciences and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [15]. In this paper, we restrict our attention to oscillation of a class of second-order differential equations of the form 1.1) rt) xt) + pt)x α τt))) ) + qt)xσt)) = 0, t t0. Throughout, we use the notation zt) := xt) + pt)x α τt)) and always assume that the following assumptions are satisfied. A 1 ) 0 < α 1 is a ratio of odd positive integers; A 2 ) r C 1 [t 0, ), 0, )), p, q C[t 0, ), [0, )), and q is not eventually zero on any half line [t, ) for t t 0 ; A 3 ) τ C[t 0, ), R), σ C 1 [t 0, ), R), τt) t, σt) t, σ t) > 0, and lim t τt) = lim t σt) =. We consider only those solutions x of 1.1) which satisfy sup{ xt) : t T } > 0 for all T t 0 and assume tha.1) possesses such solutions. As usual, a solution of 1.1) is called oscillatory if it has arbitrarily large zeros on [t 0, ); otherwise, it is called nonoscillatory. Equation 1.1) is said to be oscillatory if all its solutions are oscillatory. In the last few years, there has been much research activity concerning oscillatory behavior of various classes of differential equations. We refer the reader to [1 14, 16 32] and the references cited therein. For oscillation of second-order neutral differential equations, Agarwal et al. [1 5], Baculíková and Dˇzurina [9], Grace and Lalli [14], Han et al. [16], Hasanbulli and Rogovchenko [17], Karpuz et al. [18], Li et al. [20 24, 26], and Zafer [28] studied 1.1) when α = 1. Lin and Tang [27] considered a first-order neutral differential equation with a superlinear neutral term m [xt) px α t τ)] + qt) xt σ j ) βj sgn[xt σ j )] = 0, α > 1. j=1 Received: ; In revised form: ; Accepted: athematics Subject Classification. 34C10, 34K11. Key words and phrases. Oscillatory behavior, neutral differential equation, second-order. Corresponding author: Tongxing Li; litongx2007@163.com 1

2 2 R. P. Agarwal,. Bohner, T. Li and C. Zhang As yet, there are few results for oscillation of 1.1) in the case α 1. Hence, we will investigate equation 1.1) under the assumption that 0 < α 1. In Section 2, we consider two cases dt 1.2) rt) = and 1.3) t 0 t 0 dt rt) <. 2. AIN RESULTS In what follows, all functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough. Without loss of generality, we can deal only with the positive solutions of 1.1) in the proofs of our main theorems. Theorem 2.1. Assume A 1 ) A 3 ) and 1.2). If there exists a positive function ρ C 1 [t 0, ), R) such that [ 2.4) lim sup 1 α2 1 α + 21 α 1 pσs)) ρs)qs) t t 0 rσs))ρ s)) 2 ] 4ρs)σ ds = s) holds for all constants > 0, then 1.1) is oscillatory. Proof. Suppose to the contrary that equation 1.1) has an eventually positive solution x, i.e., there exists a t 0 such that xt) > 0, xτt)) > 0, and xσt)) > 0 for t. From 1.1), we have 2.5) rz ) t) = qt)xσt)) 0. By condition 1.2), we see that z > 0. It follows from the definition of z, [7, Lemma 2.2], and Bernoulli inequality that 2.6) xt) = zt) pt)x α τt)) = zt) pt)1 + x α τt))) + pt) In view of 2.5) and 2.6), we have zt) 2 1 α pt)1 + xτt))) α + pt) zt) 2 1 α pt)1 + αxτt))) + pt) zt) α2 1 α pt)zτt)) α )pt) 1 α2 1 α pt))zt) α )pt). 2.7) rz ) t) 1 α2 1 α pσt)))qt)zσt)) α )qt)pσt)). Define the function 2.8) ωt) := ρt) rt)z t) zσt)), t. Then ωt) > 0 for t and 2.9) ω t) = ρ t) rt)z t) zσt)) + ) t) ρt)rz zσt)) t)z σt))σ t) ρt)rt)z z 2. σt))

3 By 2.5) and σt) t, we get 2.10) rt)z t) rσt))z σt)). Neutral differential equations 3 From z > 0, there exists a constant > 0 such that zt) for all t large enough. It follows from 2.7), 2.8), 2.9), and 2.10) that ω t) 1 α2 1 α + 21 α 1 pσt)) ρt)qt) + ρ t) ρt) ωt) σ t)ω 2 t) rσt))ρt). Thus, we obtain ω t) 1 α2 1 α + 21 α 1 pσt)) ρt)qt) + rσt))ρ t)) 2 4ρt)σ. t) Integrating the last inequality from to t, we have [ 1 α2 1 α + 21 α 1 pσs)) ρs)qs) rσs))ρ s)) 2 ] 4ρs)σ ds ω ), s) which contradicts 2.4). This completes the proof. Next we establish an oscillation criterion for 1.1) in the case where 1.3) holds. Theorem 2.2. Let A 1 ) A 3 ), 1.3), and 1 α2 1 α pt)δτt))/δt) > 0 hold. Assume there exists a positive function ρ C 1 [t 0, ), R) such that 2.4) holds for all constants > 0. If 2.11) lim sup t t 0 [ 1 α δτσs))) 1 α α 1 pσs)) δσs)) Kδs) holds for all constants K > 0, where then 1.1) is oscillatory. δt) := t ds rs), 1 qs)δs) 4rs)δs) ] ds = Proof. Suppose to the contrary that equation 1.1) has an eventually positive solution x, i.e., there exists a t 0 such that xt) > 0, xτt)) > 0, and xσt)) > 0 for t. From 1.1), we have that 2.5) holds. Then we get either z > 0 or z < 0 eventually. Assume z > 0. Proceeding as in the proof of Theorem 2.1, we can get a contradiction to 2.4). Suppose z < 0. Define the function u by 2.12) ut) := rt)z t), t. zt) Then ut) < 0 for t. From 2.5), we have Hence we get which implies that 2.13) z s) rt) rs) z t), s t. l zl) zt) rt)z ds t) t rs), rt)z t) δt) 1, zt)

4 4 R. P. Agarwal,. Bohner, T. Li and C. Zhang that is, 2.14) ut)δt) 1. On the other hand, we get by 2.13) that z ) 2.15) t) 0. δ It follows from the definition of z, 2.15), [7, Lemma 2.2], and Bernoulli inequality that 2.16) xt) = zt) pt)x α τt)) = zt) pt)1 + x α τt))) + pt) zt) 2 1 α pt)1 + xτt))) α + pt) zt) 2 1 α pt)1 + αxτt))) + pt) zt) α2 1 α pt)zτt)) α )pt) 1 α2 1 α pt) δτt) zt) α )pt). δt) By virtue of 2.5) and 2.16), we have 2.17) rz ) t) 1 α2 1 α pσt)) δτσt)) qt)zσt)) α )qt)pσt)). δσt)) Differentiating 2.12), we obtain 2.18) u t) = rz ) t) zt) Combining 2.15), 2.17), and 2.18), we get 2.19) u t) 1 u2 t) rt). 1 α δτσt))) α α 1 δσt)) Kδt) pσt)) qt) u2 t) rt) for some constant K > 0. ultiplying 2.19) by δt) and integrating the resulting inequality from to t, we see that 1 α δτσs))) δt)ut) δ )u ) + 1 α α 1 pσs)) qs)δs)ds δσs)) Kδs) which yields [ 1 + us) t rs) ds + 1 α δτσs))) α α 1 δσs)) Kδs) u 2 s) δs)ds 0, rs) pσs)) 1 qs)δs) 4rs)δs) when using 2.14), this contradicts 2.11). The proof is complete. 3. EXAPLES AND DISCUSSIONS In the following, we illustrate possible applications with two examples. Example 3.1. Consider a second-order neutral differential equation ) 3.20) t xt) + 1 τt))) t xα + λxσt)) = 0, t 1, ] ds 1+δ )u ) where λ > 0 is a constant. Set ρt) = 1. Then 3.20) is oscillatory by Theorem 2.1.

5 Neutral differential equations 5 Example 3.2. Consider a second-order neutral differential equation 3.21) t 2 xt) + 1 ) ) t t t 2)) 2 xα + λx = 0, t 1, 2 where λ > 0 is a constant. Let ρt) = 1. Then 3.21) is oscillatory by Theorem 2.2. Remark 3.1. Using the Riccati transformation technique and some inequalities, two sufficient conditions are presented that can be used in oscillation problem of neutral differential equation 1.1) under the assumption that 0 < α 1. Note that when α = 1, Theorem 2.2 obtained improves [16, Theorem 2.1, Theorem 2.2, Theorem 3.1, and Theorem 3.2] in the sense that we remove the restrictive conditions that p 0, τt) = t τ 0 t, σt) t τ 0, and τ σ = σ τ, where τ 0 0 is a constant. One can easily see that results reported in [1 5,9,14,16 18,20 24,26 28] cannot be applied in equations 3.20) and 3.21) when α < 1. However, to achieve these results for the case α < 1, we are forced to impose some assumptions on the coefficient p; e.g., lim t pt) = 0. It would be interesting to find other methods to investigate 1.1) that can deal with different p. Acknowledgements. This research is supported by NNSF of P. R. China Grant Nos , , ). REFERENCES [1] Agarwal, R. P., Bohner,. and Li, W. T., Nonoscillation and Oscillation: Theory for Functional Differential Equations, volume 267 of onographs and Textbooks in Pure and Applied athematics, arcel Dekker Inc., New York, 2004 [2] Agarwal, R. P. and Grace, S. R., Oscillation theorems for certain neutral functional differential equations, Comput. ath. Appl., ), 1 11 [3] Agarwal, R. P., Grace, S. R. and O Regan, D., Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000 [4] Agarwal, R. P., Grace, S. R. and O Regan, D., Oscillation criteria for certain nth order differential equations with deviating arguments, J. ath. Anal. Appl., ), [5] Agarwal, R. P., Grace, S. R. and O Regan, D., The oscillation of certain higher-order functional differential equations, ath. Comput. odelling, ), [6] Agarwal, R. P., Shieh, S. L. and Yeh, C. C., Oscillation criteria for second-order retarded differential equations, ath. Comput. odelling, ), l 11 [7] Baculíková, B., Properties of third-order nonlinear functional differential equations with mixed arguments, Abstr. Appl. Anal., ), 1 15 [8] Baculíková, B., Oscillation criteria for second order nonlinear differential equations, Arch. ath., ), [9] Baculíková, B. and Džurina, J., Oscillation theorems for second order neutral differential equations, Comput. ath. Appl., ), [10] Dˇzurina, J., Oscillation theorems for neutral differential equations of higher order, Czech. ath. J., ), [11] Dˇzurina, J. and Stavroulakis, I. P., Oscillation criteria for second order delay differential equations, Appl. ath. Comput., ), [12] Erbe, L., Kong, Q. K. and Zhang, B. G., Oscillation Theory for Functional Differential Equations, arcel Dekker Inc., New York, 1995 [13] Grace, S. R., Bohner,. and Liu, A., On Kneser solutions of third-order delay dynamic equations, Carpathian J. ath., ), [14] Grace, S. R. and Lalli, B. S., Oscillation of nonlinear second order neutral delay differential equations, Rad. ath., ), [15] Hale, J. K., Theory of Functional Differential Equations, Springer-Verlag, New York, 1977 [16] Han, Z., Li, T., Sun, S. and Sun, Y., Remarks on the paper [Appl. ath. Comput., ) ], Appl. ath. Comput., ),

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